MTH 265 Written Homework 1 - Taylor Polynomials
Instructions
Answer the following questions on a separate sheet of paper. Your final write up of solutions should
• include a paraphrase of the problem,
• show clearly organized work toward a solution, and
• conclude with sentence(s) containing your answer.
Problems
1. Consider $f(x) = \sin(x)$ and its Taylor polynomial.
$m = 1$
(a) Find the 5th degree Taylor polynomial $T_5(x)$ centered at $x = \pi$.
(b) Use the Taylor Remainder to find a bound on the error on the interval $[0, 2\pi]$. That is, according to the Taylor Remainder, how good is this approximation on the interval $[0, 2\pi]$?
(c) Graph $f(x)$ and $T_5(x)$ and find the actual error and illustrate on the graph.
(d) Consider $T_n(x)$. Use the Taylor Remainder to find a bound on the error on the interval $[0, 2\pi]$ in terms of $n$.
2. Consider $f(x) = \ln(1 + x)$ and the corresponding Taylor Polynomials centered at $x = 0$.
(a) Find a formula for $f^{(k)}(x)$ in terms of $k$. Hint: Take derivatives. Find the pattern.
(b) Use this to find $T_n(x)$ for any value of $n$.
(c) Determine the degree $n$ required for the error at $x = 0.5$ to be less than $0.0001$, and give the corresponding Taylor Polynomial.
(d) For that value of $n$, find the largest interval containing $x = 0$ over which $\ln(x + 1)$ can be approximated by $T_n(x)$ with an error of at most $0.0001$.
(e) Graph your results.
$\cos$
$\sin \quad -\sin$
$-\cos$
